Low Complexity Spatial Similarity Measure of GPS Trajectories
Radu Mariescu-Istodor, Andrei Tabarcea, Rahim Saeidi and Pasi Fr
¨
anti
Speech and Image Processing Unit, School of Computing, University of Eastern Finland, Joensuu, Finland
Keywords:
GPS Trajectory, Spatial Similarity, MGRS, Cell Approximation, Sampling Frequency, Interpolation, Dilation.
Abstract:
We attack the problem of trajectory similarity by approximating the trajectories using a geographical grid
based on the MGRS 2D coordinate system. We propose a spatial similarity measure which is computationally
feasible for big data collections. The proposed measure is based on cell matching with a similarity metric
drawn from Jaccard index. We equip the proposed method with interpolation and dilation to overcome the
problems missing data and different sampling frequencies when comparing two trajectories. The proposed
measure is implemented online in the framework of Mopsi
a
.
a
cs.uef.fi/mopsi
1 INTRODUCTION
In recent years, GPS technology has been widely
available in consumer devices, especially in smart-
phones
1
, which count as more than a half on total
mobile phone sales
2
. Furthermore, most of the users
utilize their phone to find their location, amongst
other services
3
. The wide availability of GPS-enabled
smartphones that are also connected to the Internet
has made the collection of large amount of location-
based data possible. Such data includes geo-tagged
photos, videos and geographical trajectories. Col-
lecting geographical trajectories has practical appli-
cations in fleet management, sports tracking, rec-
ommending tourist trajectories, improving navigation
and determining mobility patterns.
Having a large-scale collection of GPS trajecto-
ries raises the challenge of how to organize the data,
how to present it in a meaningful way and how to
filter out irrelevant data. Computing trajectory sim-
ilarity is a tool that can be used in addressing those
challenges (Agrawal et al., 1993). A problem in com-
puting similarity of GPS trajectories is that the large
amount of data does not permit processing raw trajec-
tories in real time.
Time series analysis of one-dimensional data
1
abiresearch.com/research/product/1005746-mobile-
device-user-interfaces
2
gartner.com/newsroom/id/2623415
3
pewinternet.org/Reports/2012/Location-based-servi
ces.aspx
across the time has been used for analyzing stock
changes, weather data and biomedical measurements
(Hamilton, 1994; Chan and Fu, 1999; Worsley and
Friston, 1995; Lange and Naumann, 2011). Despite
the significant research output on time series analy-
sis, the concept of computing similarity for traces of
moving objects in the framework of spatio-temporal
databases has been studied much less. Finding k-
nearest trajectories, indexing and clustering of spatio-
temporal data are among the recent directions of re-
search with many applications to make queries in
moving object databases (Frentzos et al., 2007a; Ni
and Ravishankar, 2007; Frentzos et al., 2007b; G
¨
uting
et al., 2010; Pelekis et al., 2011). These algorithms
can be applied also for measuring the trajectory simi-
larity (Hu and Steenkiste, 2006).
Using Euclidean distance is not practical for the
case that the length of two trajectories are not equal
(Yanagisawa et al., 2003). Dynamic time warping
handles matching two sequences of different length
but it is very sensitive to noisy data (Berndt and Clif-
ford, 1994). Algorithms like longest common subse-
quence (LCS) (Vlachos et al., 2002b; Vlachos et al.,
2002a) or edit distance on real sequence (EDR) (Chen
et al., 2005) are designed to account for noisy and
missing data but they are not perturbation free. Con-
sidering M trajectories of N points on average, the
computational complexity of these algorithms is at
minimum O(M
2
· N
2
). Hence, these algorithms can-
not provide real-time results when dealing with a
large collection of data.
These algorithms do not utilize time stamps. By
62
Mariescu-Istodor R., Tabarcea A., Saeidi R. and Fränti P..
Low Complexity Spatial Similarity Measure of GPS Trajectories.
DOI: 10.5220/0004940500620069
In Proceedings of the 10th International Conference on Web Information Systems and Technologies (WEBIST-2014), pages 62-69
ISBN: 978-989-758-023-9
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
using the timing information a complete movement
profile can be provided and the similarity of two tra-
jectories can be used in trajectory clustering applica-
tions. The similarity measurement in LCS and EDR
are based on point-to-point distance calculations. In
the event of having two trajectories with different
sampling frequency, LCS and EDR cannot provide
correct similarity measure (Frentzos et al., 2007b).
Although it is always possible to use a trajectory re-
duction or approximation algorithm to represent a tra-
jectory with far less representatives for similarity cal-
culation, the quality of such an approximation algo-
rithm and overhead computational complexity is de-
batable (Ni and Ravishankar, 2007).
In this paper, we propose a fast method of comput-
ing trajectory similarity by approximating the trajec-
tories using a geographical grid based on a 2D coor-
dinate system. This process reduces a trajectory from
points to cells with order of magnitude less details in
representation and subsequently in distance calcula-
tions. We employ an asymmetric similarity metric in-
spired by Jaccard index. Dealing with GPS data col-
lection, it is common to have bunch of data points lost
or compare trajectories traveled by car with walking
speed trajectories. We propose interpolation and di-
lation of trajectories represented as cells to overcome
these difficulties. In the results section we simulate
missing data and trajectory sampling frequency mis-
match with two example trajectories and demonstrate
the efficiency of the proposed approach. Conclusions
are drawn after the discussion of results.
2 MOPSI
Mopsi is a research project location-based service
developed at the University of Eastern Finland by
Speech and Image Processing Group from the School
of Computing. (Fr
¨
anti et al., 2011) Mopsi offers
multiple applications of location-aware systems, be-
ing a test-bed for various research topics that involve
location-aware data. It contains tools for collecting,
processing and displaying location-based data, such
as photos or trajectories, along with social media in-
tegration. The main topics addressed in Mopsi are
collecting location-based data, mining location data
from web pages, processing, storing and compress-
ing of GPS trajectories, detecting transportation mode
from GPS trajectories, recommending points of in-
terest, using location information in social networks,
detecting users actions by using their location and
building location-based games with the help of user-
generated collections.
Location-based data is very common among web-
Figure 1: Mopsi application on web showing an example of
two trajectories which display a common region.
pages, especially when their content describe com-
mercial services, landmarks or public institutions.
However, the location data is more commonly pre-
sented in a human-readable way and not as geograph-
ical coordinates, which are more accurate and easier
to be automatically identified. We propose a method
to automatically identify location information from
web-pages by detecting postal addresses (Fr
¨
anti et al.,
2010).
Mopsi provides tools to collect GPS trajectories
and it includes more than 9000 trajectories composed
of over 7 million points by the end of 2013. Mopsi
uses fast retrieval and displaying of the data (Waga
et al., 2013) based on GPS trajectory polygonal ap-
proximation (Chen et al., 2012a). GPS trajectories are
also compressed for optimizing storage space (Chen
et al., 2012b). Transport mode information can be
also retrieved by automatically analyzing GPS trajec-
tories (Waga et al., 2012). The algorithm uses a sec-
ond order Markov model to segment the trajectories
and to detect car, bicycle, running or walking trans-
portation modes.
The relevance of location-based media can be as-
sessed by considering several aspects such as time, lo-
cation, content or social network (Fr
¨
anti et al., 2011),
which are used to create a context for each user. A
personalized recommender system can recommend
relevant data based on user location and user context
(Waga et al., 2011). Such data can be geotagged pho-
tos, services confirmed by administrators or GPS tra-
jectories. Users can share their location in real-time
by using mobile phone location-aware applications.
This allows for the detection of various location-
based actions such as meetings, visiting or passing-
by points of interest (Mariescu-Istodor, 2013). Mopsi
also includes location-based games, such as O-Mopsi
(Tabarcea et al., 2013), which is an orienteering game
using the data from a user-generated photo collection.
Mopsi provides tools for collecting location-based
LowComplexitySpatialSimilarityMeasureofGPSTrajectories
63
Figure 2: MGRS grid zones (source
4
).
data with mobile devices. It is available on most
mobile operating systems (Android, iOS, Windows
Phone, Symbian). The server-side processes the data
collected by the user and displays the data collec-
tion. It also provides social features and integration
which social media, with functionalities such as chat-
ting, friends tracking and sharing data to Facebook.
The Mopsi routes module provides tools for trajectory
recording and displaying the large amount of data in
reasonable time. Trajectory similarity is the newest
addition to the Mopsi routes module.
3 TRAJECTORIES
In Mopsi we record a user’s location at a certain time
as a point p
k
= (x
k
,y
k
,t
k
), where x
k
is the latitude,
y
k
is the longitude and t
k
is the timestamp of point k.
An ordered sequence of these points, defines a spatial
trajectory R = (p
1
,...,p
K
). We calculate the similar-
ity between a reference trajectory R
a
and all the other
M 1 trajectories in the database, R
m
, m = 1, . . . , M.
The similarity of two trajectories can be calculated
as the Jaccard index:
J(R
a
,R
m
) =
|R
a
R
m
|
|R
a
R
m
|
, (1)
Instead of this symmetric measure we want to find
out if the reference trajectory is completely covered
by another trajectory. Thus, we consider the follow-
ing asymmetric similarity metric:
Sim(R
a
,R
m
) =
|R
a
R
m
|
|R
a
|
, (2)
Sim(R
m
,R
a
) =
|R
a
R
m
|
|R
m
|
. (3)
4
earth-info.nga.mil/GandG/coordsys/grids/universal grid syst
em.html
The first one shows what percentage of R
a
is shared
with R
m
and the second shows what percentage of R
m
is shared by R
a
. The way that we perform intersection
operator is described in the following sections after
we quantize the trajectories into cells.
3.1 Cell Approximation
In a preprocessing step, we generate a cell repre-
sentation for a trajectory after it has been recorded.
The Military Grid Reference Systems (MGRS) is an
alpha-numeric system for expressing UTM/UPS co-
ordinates. MGRS is used by NATO to locate points on
earth. A single alpha-numeric value references a po-
sition that is unique for the entire earth (see Figure 2).
MGRS is a projected coordinate system which uses a
2-dimensional Cartesian horizontal position orienta-
tion, so that locations are identified independently of
vertical position. MGRS shares several characteris-
tics with UTM such as the division of earth into pro-
jection zones and using easting and northing in meters
within a designated zone. The main differences are
that a MGRS zone is a 100km square within a UTM
zone, whilst a UTM zone is usually 6 degrees in east-
west and 8 degrees in north-south area and also that
the notation of the areas is different. Based on the
coordinate resolution, MGRS can define a grid with
square cells with the length starting from 100km up
to 10m or even 1m.
We approximate a trajectory R = {(x
k
,y
k
)}
K
k=1
by
a sparse binary matrix representation C where,
(C)
i j
=
(
1 0 < x
k
iL < L, 0 < y
k
jL < L
0 Otherwise
, (4)
where L stands for the cell length (25 meters in this
paper) and indexes i and j span over in horizontal and
vertical cells that trajectory R is residing inside. Fig-
ure 3 shows how the reference trajectory is approxi-
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64
Figure 3: Example of a trajectory of 420 points being rep-
resented by 35 cells using the approximation in Equation 4.
The cell representation is not continuous. The gaps appear
because of the fixed cell size, variations in movement speed
(or different sampling frequencies) and missing GPS loca-
tions. It is likely for such gaps to appear especially when
users are moving by car, train or plane.
mated by cells. Generating the cell representation for
a trajectory of average length of N points is done in
O(N) time.
3.2 Measuring Similarity
The similarity between two trajectories R
a
,R
m
can
now be calculated as:
Sim(R
a
,R
m
) =
kC
a
C
m
k
0
kC
a
k
0
, (5)
where C
a
and C
m
are the cell representations of R
a
and R
m
, respectively, C
a
C
m
is a Hadamard product
of two matrices C
a
and C
m
defined as (C
a
C
m
)
i j
=
(C
a
)
i j
·(C
m
)
i j
and kCk
0
represents the `
0
-quasinorm.
In implementation, C
a
and C
m
are multiplied element
by element and then we measure the number of non-
zero elements. Figure 4 shows two sample trajectories
being matched.
Assuming we have the cell representation C of a
reference trajectory R we calculate the similarity for
all trajectories in database in two steps. First, we find
all the trajectories which share at least one cell with
the reference trajectory. This has a time complexity of
O(N
0
· (q + M
0
)) where q represents the steps needed
by the database system to perform the search (N
0
N
and M
0
M). In contrast to the average length N of
a trajectory R, we define N
0
= kCk
0
as the number
of non-zero elements in cell-approximated version of
R. In a similar way, M
0
indicates the number of other
trajectories that share at least one cell with trajectory
R. Secondly, we calculate the trajectory similarity
according to Equation (5) with a time complexity of
O(M
0
· N
0
). The overall complexity of the similarity
Figure 4: Matching two trajectories using the cell repre-
sentation. The green cells denote the reference trajectory
and the gray cells represent the other trajectory. The ’x’
symbol is used to mark the cells shared by two trajectories;
Sim(C
a
,C
m
) = 40% and Sim(C
m
,C
a
) = 31%
.
scoring is O(M
0
· N
0
) (assuming q constant by adding
a proper indexing structure in the database).
In Figure 4 the straighforward application of the
similarity scores yield similarity scores of 40% and
31% even though the trajectories seem to have more
than 50% similarity by visual inspection. In the next
subsections we analyze why this happens.
3.3 Interpolation
When the user is traveling fast or when recording fre-
quency is low we notice gaps in the trajectory repre-
sentation by cells. Gaps can also appear due to lack of
GPS signal. Figure 5 shows three examples when dif-
ferent sized gaps appear in the cell representation of a
trajectory. In cell approximation stage in Equation 4,
we process the trajectory data points in the sequence
they are recorded. In this way, the sequence of cells
being detected as “1s” are used to determine if the
next cell is connected to the current cell and find a
potential gap in cell-approximation.
In order to fill the gap, the line equation between
two cells is obtained from the start and end points as
j = f (i) =
j
2
j
1
i
2
i
1
(i i
1
) + j
1
(6)
where i
1
and j
1
are the coordinates of one cell and
i
2
, j
2
are the coordinates of the other cell. The line
in Equation 6 is then sampled by the cells that it is
passing through and then set respective cell values as
(C
i j
) = 1.
By performing interpolation, the trajectory simi-
larity presented in Figure 4 is now updated as plotted
in Figure 6. The similarity values are still below the
visual expectations. The reason is that two cell repre-
sentations may not overlap even though the trajecto-
ries are close to each other.
LowComplexitySpatialSimilarityMeasureofGPSTrajectories
65
Figure 5: Interpolation between two cells in order to fill a
gap; three example situations are depicted.
Figure 6: The trajectory having gaps is interpolated and the
matching of the two trajectories becomes: Sim(C
a
,C
m
) =
41% and Sim(C
m
,C
a
) = 33%.
3.4 Dilation
A frequent situation is that two nearby trajectory seg-
ments are evolving along each other in cell represen-
tation instead of overlapping. An example is provided
in Figure 7 We solve this issue by applying morpho-
logical dilation on the trajectories and taking into ac-
count the neighbouring cells of a trajectory. We define
C
d
as a result of binary dilation of sparse binary rep-
resentation C by binary structure S with
C
d
= C S = T (C S), (7)
where defines the binary dilation and indicates the
convolution operator. In the Equation 7, T (·) stands
for binarization transform as
T ((C S)
i j
) =
(
0 0 (C S)
i j
< 1
1 Otherwise
(8)
Figure 7: We see that two trajectories which are close
enough to be considered similar can be represented by dif-
ferent cells. Only a single cell is shared by the cell repre-
sentation of the two trajectories.
Figure 8: The reference trajectory is dilated and the match-
ing of the two trajectories becomes: Sim(C
a
,C
m
) = 64%
and Sim(C
m
,C
a
) = 53%.
Figure 8 shows how a trajectory is dilated with the
following structure
S =
1 1 1
1 1 1
1 1 1
. (9)
Then the two trajectories are matched when one of
the trajectories is dilated. The similarity score is now
calculated with C
a
and C
d
m
as in Equation 5. Typi-
cally the number of cells used in the trajectory repre-
sentations increases by a factor of 3 when dilation is
applied.
4 RESULTS
We implement our method in a real-world application,
as a prototype using the Mopsi project route analy-
sis module
5
. We investigate issues that may appear
5
cs.uef.fi/mopsi/?tab=routes&userId=13&routeId=137882401
9381&similarity=true
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66
No post-processing Interpolated Interpolated and Dialated
5x
29%
31%
53%
10x
24%
31%
53%
15x
18%
27%
53%
Figure 9: Simulating different sampling frequencies by subsampling the reference trajectory with a factor of 5x, 10x and 15x.
when collecting GPS trajectories in a practical appli-
cation such as different sample rates, interpolation of
collected points or breaks in the GPS signal caused by
technical or environmental problems.
Firstly, as shown in Figure 9, we investigate how
a different sampling frequency impacts the similarity
score calculation. The reference trajectory is subsam-
pled with factor f by only keeping every f
th
element
from the original trajectory. We notice that the in-
terpolation step doesn’t increase the similarity scores
significantly. However, when followed by dilation,
the similarity score indicates robustness against vari-
ations in sampling frequency which is a desired prop-
erty for a trajectory matching procedure.
The other common issue while recording a trajec-
tory is loss of location information for a brief period
of time. This can happen, for example, if the user
goes through a building, a tunnel or simply due to de-
vice software error. We simulate this behavior and
see how the similarity scoring is affected in Figure 10.
When removing 90 points we notice that the similar-
ity score has dropped even when using interpolation
and dilation. This happened because we removed a
significant amount of subsequent points (20% of the
trajectory). Interpolation does not have enough infor-
mation to reconstruct the trajectory appropriately and
consequently, loss of many data points in a trajectory
is detrimental for similarity calculations.
The proposed method is implemented in two steps
for real-world application: the preprocessing step,
LowComplexitySpatialSimilarityMeasureofGPSTrajectories
67
No post-processing Interpolated Interpolated and Dialated
(-50)
20%
31%
53%
(-90)
22%
27%
42%
Figure 10: Simulating loss of GPS signal by removing 50 and 90 sequential points from the reference trajectory.
done when a new trajectory is added into the sys-
tem and the similarity score calculation step, per-
formed when searching all the similar trajectories of a
given trajectory. When not using interpolation or di-
lation the time complexity for the preprocessing step
is O(M · N) for M trajectories of average length N
points. The similarity score calculation has a time
complexity of O(M
0
· N
0
). After interpolation is ap-
plied there will be an increase on the N
0
and M
0
pa-
rameters which increase, however, stay at the same
order of magnitude. N
0
increases by the number of
cells added trough interpolation and M
0
increases by
the number of trajectories that share at least one cell
with interpolated trajectory. The dilation stage in-
creases the N
0
and M
0
parameters once more. N
0
typ-
ically increases by a factor of 3 and M
0
grows by
the number of trajectories that share the cells that
are added to the representation as a result of dila-
tion. The overall complexity for M trajectories in the
database is governed by O(M · N) for cell approxi-
mation and O(α · M · M
0
· N
0
) for similarity score cal-
culation including interpolation and dilation (α 6,
M
0
M, N
0
N). The similarity cell approxima-
tion complexity of O(M · N) is negligible compared
to O(α · M · M
0
· N
0
) for score calculation. Hence, the
overall computational complexity of the proposed ap-
proach is dominated by O(α · M · M
0
· N
0
) which is
comparably much less than O(M
2
· N
2
) for other sim-
ilarity metrics presented in section 1.
5 CONCLUSIONS
We presented a method for computing similarity be-
tween trajectories in a large data collection. Because
trajectories are likely to have different speed profile
and missing points, interpolation and dilation tech-
niques are employed before the scoring. We have
demonstrated that the method is robust except when
many points are removed and dramatically affect the
structure of a trajectory. In that situation there is sim-
ply not enough information to rebuild the path and
provide correct similarity values. The method was
implemented in Mopsi, where for a given trajectory
we display a list of similar paths in reverse order of
the similarity scores.
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